I've had Dave's Short Course on Complex Numbers on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers. I'm frequently asked that question.
This is not a question about the existence or usefulness of complex numbers as they do exist as mathematical entities and they are very useful. It is a question about what makes them numbers. It could be broadened to ask what constitutes numbers, but I'm specifically interested in complex numbers.
Perhaps it's just historical and sociological, but I'm hoping for an answer that there's something intrinsic about complex numbers that makes them fit into the category of number.
Best Answer
Complex numbers are a hack. That is, a retrofit. They were discovered, not planned for, and added to our system of math.
The same is true for imaginary, transcendental, real, rational, and negative numbers. Zero and infinity as well.
They are all additions to the counting / natural numbers. In different ways, they all work by rejecting the premise of a question. That is, rejecting its assumption.
Negative
How many apples will you owe me if I give you this apple? -1. Minus one? No, negative one. You already owe me two from yesterday. So right now I owe you -2. Negatives reject an assumption by completely reversing it. This could have been avoided if the question had assumed the speaker was the one in debt. By making the subtraction operation part of a number we don't have to force the question to change to answer it.
Rational
How many apples will you and I get if Mom gives us both 1 apple? 1/2 apple. What's a 1/2 apple? Cut an apple in two. Each piece is a 1/2 apple. Here we're making division part of a number so we can answer the question with counting numbers. This rejects the questions assumption that we will be getting whole apples. This could have been avoided if the question had asked about parts of apples. Making the division operation part of the number lets us answer the question without forcing the question to change.
Irrational (but not transcendental)
If I walk diagonally rather than only in North, South, East, West directions how much can that reduce the distance I have to walk? At most about 0.70710678118654752440084436210485 (a savings of almost 30%) That's a lot of digits for an about. Can't you be exact? Sure, it's exactly √2/2.
Here we're rejecting the assumption that a perfect answer is expressible in counting numbers with anything less than an infinite number of divisions. Instead we make a root operation part of the number to solve a polynomial problem.
Transcendental
How many 10 foot panels should I buy to enclose a circular coral around a horse run with a 12 foot lead line? At least 8. At least? Well if you could buy curved panels and get at least one at a custom length and you didn't mind the horse having to run on top of it then the answer is about 7.5398223686155037723103441198708 ten foot panels. Again with the digits! OK OK, it's exactly (2π)(12/10). Transcendental numbers reject the assumption that a perfect answer is expressible in counting numbers, even with polynomial operations, without symbolic constants like π. (I know where the 12 and the 10 come from, but why the 2? Tradition.)
Real
How many inches tall are you? 71.5 inches. 71 and what? .5, take an inch, divide it into 10 pieces of inch, 5 of those pieces plus 71 inches is how tall I am. This rejects the assumption that you can measure my height with whole inches. Again, we're making the division operation part of a number. But now it's an arbitrary number of division operations. Allow that go on for up to an infinite number of divisions and you get the real number line.
Imaginary and Complex
What number can I multiply by itself and get a negative 5? You can't. What if I could? Huh? Imagine I can. What would that be? Imaginary numbers reject the assumption that numerical answers to a question must either exist on the number line, as conceived by the question, or not exist at all. Much like the other extensions to the counting numbers I've mentioned here imaginary numbers hide an operation, the square root of -1, within them letting you procrastinate dealing with it and it's consequences. Add any non zero real number to that imaginary number and it's suddenly complex. We can't really add it so we just leave it. Procrastinating again.
Zero and Infinity
How many days are left in February? Zero, it's March already. Zero is the most iconic rejection of a question's assumption. It's a relatively new idea. Some number systems don't even have it (for example, roman numerals).
How fast will I have to drive to get to Albany when you do? Infinite, I just arrived. Infinity also rejects an assumption, be it a limit or a rate.
Both zero and infinity are some of the poorest behaved numbers in our system of math. They are very useful but they are destroyers of information if not guarded against. Let them loose unaware and they can convince you that 1 = 2.
Matrices
The biggest thing complex numbers have going for them is that unlike matrices they work in our algebra. Infact you can think of the preceding numbers as nothing but a list of things we can cram into the system without breaking it.
Matrix multiplication is not commutative. This breaks the system. Now when you deal with X you have to KNOW whether or not X is a matrix or you don't know what is legal to do with it. Of the numbers mentioned so far the only ones that come close to this problem are zero and infinity. So I would argue that complex numbers have more of a right to be called numbers than zero does.
Numbers
Thought of this way, a number could be any mathematical expression, free of unknowns, that behaves reasonably well within our algebra. If I can stick it in X and forget what it is for a time, I think that's a number.